Inverse limits and statistical properties for chaotic implicitly defined economic models

Abstract

In this paper we study the dynamics and ergodic theory of certain economic models which are implicitly defined. We consider 1-dimensional and 2-dimensional overlapping generations models, a cash-in-advance model, heterogeneous markets and a cobweb model with adaptive adjustment. We consider the inverse limit spaces of certain chaotic invariant fractal sets and their metric, ergodic and stability properties. The inverse limits give the set of intertemporal perfect foresight equilibria for the economic problem considered. First we show that the inverse limits of these models are stable under perturbations. We prove that the inverse limits are expansive and have specification property. We then employ utility functions on inverse limits in our case. We give two ways to rank such utility functions. First, when perturbing certain dynamical systems, we rank utility functions in terms of their average values with respect to invariant probability measures on inverse limits, especially with respect to measures of maximal entropy. For families of certain unimodal maps we can adjust both the discount factor and the system parameters in order to obtain maximal average value of the utility. The second way to rank utility functions (for more general maps on hyperbolic sets) will be to use equilibrium measures of these utility functions on inverse limits; they optimize average values of utility functions while at the same time keeping the disorder in the system as low as possible in the long run.

Refinement of dynamic equilibrium using small random perturbations

“…If, in addition, that bpf map exhibits chaotic behavior, Gardini et al (2009) proved that the set of forward paths that are intertemporal equilibria has a fractal attractor. Characterizations and properties of those intertemporal perfect foresight equilibria were studied using the inverse limit approach by Medio and Raines (2007) and Mihailescu (2012).…”

Generating multi-wing hidden attractors with only stable node-foci via non-autonomous approach

“…In recent decades, the study of nonlinear dynamics and chaos has attracted much attention [1][2][3][4]. Complex chaotic behaviors have been validated existing in various dynamical systems [5][6][7][8][9][10].…”

Complex Dynamics in Non-linear Real Estate Models

“…He investigated chaotic attractors, repellers and general hyperbolic sets and he characterized folded attractors and repellers as those basic sets with full stable and unstable dimension respectively. Also Mihailescu [47] studied the dynamics and ergodic theory of several economic models, including the cobweb model with adaptive adjustment. He considered inverse limit spaces of certain chaotic invariant fractal sets and their metric, ergodic and stability properties.…”